Stochastic simulations of a multi-group compartmental model for Johne's disease on US dairy herds with test-based culling intervention

Infection elimination may be an important goal of control programs. Only in stochastic infection models can true infection elimination be observed as a fadeout. The phenomena of fadeout and variable prevalence are important in understanding the transmission dynamics of infectious diseases and these phenomena are essential to evaluate the effectiveness of control measures. To investigate the stochastic dynamics of Mycobacterium avium subsp. paratuberculosis (MAP) infection on US dairy herds with test-based culling intervention, we developed a multi-group stochastic compartmental model (a continuous time Markov chain model) with both horizontal and vertical transmission. The stochastic model predicted fadeout and within-herd prevalence to have a large variance. Although test-based culling intervention generally decreased prevalence over time, it took longer than desired by producers to eliminate the endemic MAP infection from a herd. Uncertainty analysis showed that, using annual culture test and culling of only high shedders or culling of both low and high shedders with a 12-month delay in culling of low shedders, MAP infection persisted in many herds beyond 20 years. While using semi-annual culture test and culling of low and high shedders with a 6-month delay in culling of low shedders, MAP infection in many herds would be extinct within 20 years. Sensitivity analysis of the cumulative density function of fadeout suggested that combining test-based culling intervention and reduction of transmission rates through improved management between susceptible calves and shedding animals may be more effective than either alone in eliminating endemic MAP infection. We also discussed the effects of other factors such as herd size, heifer replacement, and adult cow infection on the probability of fadeout.     

Analysis and simulation of a stochastic, discrete-individual model of STD transmission with partnership concurrency

Deterministic differential equation models indicate that partnership concurrency and non-homogeneous mixing patterns play an important role in the spread of sexually transmitted infections. Stochastic discrete-individual simulation studies arrive at similar conclusions, but from a very different modeling perspective. This paper presents a stochastic discrete-individual infection model that helps to unify these two approaches to infection modeling. The model allows for both partnership concurrency, as well as the infection, recovery, and reinfection of an individual from repeated contact with a partner, as occurs with many mucosal infections. The simplest form of the model is a network-valued Markov chain, where the network's nodes are individuals and arcs represent partnerships. Connections between the differential equation and discrete-individual approaches are constructed with large-population limits that approximate endemic levels and equilibrium probability distributions that describe partnership concurrency. A more general form of the discrete-individual model that allows for semi-Markovian dynamics and heterogeneous contact patterns is implemented in simulation software. Analytical and simulation results indicate that the basic reproduction number R(0) increases when reinfection is possible, and the epidemic rate of rise and endemic levels are not related by 1-1/R(0), when partnerships are not point-time processes.     

Moment Equations and Dynamics of a Household SIS Epidemiological Model

An SIS epidemiological model of individuals partitioned into households is studied, where infections take place either within or between households, the latter generally happening much less frequently. The model is explored using stochastic spatial simulations, as well as mathematical models which consist of an infinite system of ordinary differential equations for the moments of the distribution describing the proportions of individuals who are infectious among households. Various moment-closure approximations are used to truncate the system of ODEs to finite systems of equations. These approximations can sometimes lead to a system of ill-behaved ODEs which predict moments which become negative or unbounded. A reparametrization of the ODEs is then developed, which forces all moments to satisfy necessary constraints.Changing the proportion of contacts within and between households does not change the endemic equilibrium, but does affect the amount of time it takes to approach the fixed point; increasing the proportion of contacts within households slows the spread of the infection toward endemic equilibrium. The system of moment equations does describe this phenomenon, although less accurately in the limit as the proportion of between-household contacts approaches zero. The results indicate that although controlling the movement of individuals does not affect the long-term frequency of an infection with SIS dynamics, it can have a large effect on the time-scale of the dynamics, which may provide an opportunity for other controls such as immunizations to be applied.     

Epidemiological models for sexually transmitted diseases

The classical models for sexually transmitted infections assume homogeneous mixing either between all males and females or between certain subgroups of males and females with heterogeneous contact rates. This implies that everybody is all the time at risk of acquiring an infection. These models ignore the fact that the formation of a pair of two susceptibles renders them in a sense temporarily immune to infection as long as the partners do not separate and have no contacts with other partners. The present paper takes into account the phenomenon of pair formation by introducing explicitly a pairing rate and a separation rate. The infection transmission dynamics depends on the contact rate within a pair and the duration of a partnership. It turns out that endemic equilibria can only exist if the separation rate is sufficiently large in order to ensure the necessary number of sexual partners. The classical models are recovered if one lets the separation rate tend to infinity.This work has been supported by Deutsche Forschungsgemeinschaft     

Stochastic modelling of environmental variation for biological populations

We examine stochastic effects, in particular environmental variability, in population models of biological systems. Some simple models of environmental stochasticity are suggested, and we demonstrate a number of analytic approximations and simulation-based approaches that can usefully be applied to them. Initially, these techniques, including moment-closure approximations and local linearization, are explored in the context of a simple and relatively tractable process. Our presentation seeks to introduce these techniques to a broad-based audience of applied modellers. Therefore, as a test case, we study a natural stochastic formulation of a non-linear deterministic model for nematode infections in ruminants, proposed by Roberts and Grenfell (1991). This system is particularly suitable for our purposes, since it captures the essence of more complicated formulations of parasite demography and herd immunity found in the literature. We explore two modes of behaviour. In the endemic regime the stochastic dynamic fluctuates widely around the non-zero fixed points of the deterministic model. Enhancement of these fluctuations in the presence of environmental stochasticity can lead to extinction events. Using a simple model of environmental fluctuations we show that the magnitude of this system response reflects not only the variance of environmental noise, but also its autocorrelation structure. In the managed regime host-replacement is modelled via periodic perturbation of the population variables. In the absence of environmental variation stochastic effects are negligible, and we examine the system response to a realistic environmental perturbation based on the effect of micro-climatic fluctuations on the contact rate. The resultant stochastic effects and the relevance of analytic approximations based on simple models of environmental stochasticity are discussed.     

A Bayesian approach for inferring the dynamics of partially observed endemic infectious diseases from space-time-genetic data

We describe a statistical framework for reconstructing the sequence of transmission events between observed cases of an endemic infectious disease using genetic, temporal and spatial information. Previous approaches to reconstructing transmission trees have assumed all infections in the study area originated from a single introduction and that a large fraction of cases were observed. There are as yet no approaches appropriate for endemic situations in which a disease is already well established in a host population and in which there may be multiple origins of infection, or that can enumerate unobserved infections missing from the sample. Our proposed framework addresses these shortcomings, enabling reconstruction of partially observed transmission trees and estimating the number of cases missing from the sample. Analyses of simulated datasets show the method to be accurate in identifying direct transmissions, while introductions and transmissions via one or more unsampled intermediate cases could be identified at high to moderate levels of case detection. When applied to partial genome sequences of rabies virus sampled from an endemic region of South Africa, our method reveals several distinct transmission cycles with little contact between them, and direct transmission over long distances suggesting significant anthropogenic influence in the movement of infected dogs.     

Mathematical modelling of immunity to malaria

A comparison of two epidemiological models of immunity to malaria shows that different characterizations of immunity boosted by exposure to infection generate qualitatively different results. Attempts to control disease by reducing transmission or increasing the recovery rate can produce an increase in prevalence in the compartmental model with discrete epidemiological states. However, the parasite density always decreases in response to disease control in the model with continuous epidemiological variables. Each model accounts for some epidemiological patterns. The increase in prevalence seen in the compartmental model is in accord with observed effects of variation in transmission. Parasite suppression in areas of antimalarial drug use is consistent with the effect of an increased recovery rate in the density model. Future work should combine the two approaches, perhaps by using the compartmental model over the low to moderate range of infection rates and switching to the density model at high infection rates. In any case, the validation of models needs to take account of the usage of antimalarial drugs as well as the intensity of transmission.     

Serotype-Specific Transmission and Waning Immunity of Endemic Foot-and-Mouth Disease Virus in Cameroon

Foot-and-mouth disease virus (FMDV) causes morbidity and mortality in a range of animals and threatens local economies by acting as a barrier to international trade. The outbreak in the United Kingdom in 2001 that cost billions to control highlighted the risk that the pathogen poses to agriculture. In response, several mathematical models have been developed to parameterize and predict both transmission dynamics and optimal disease control. However, a lack of understanding of the multi-strain etiology prevents characterization of multi-strain dynamics. Here, we use data from FMDV serology in an endemic setting to probe strain-specific transmission and immunodynamics. Five serotypes of FMDV affect cattle in the Far North Region of Cameroon. We fit both catalytic and reverse catalytic models to serological data to estimate the force of infection and the rate of waning immunity, and to detect periods of sustained transmission. For serotypes SAT2, SAT3, and type A, a model assuming life-long immunity fit better. For serotypes SAT1 and type O, the better-fit model suggests that immunity may wane over time. Our analysis further indicates that type O has the greatest force of infection and the longest duration of immunity. Estimates for the force of infection were time-varying and indicated that serotypes SAT1 and O displayed endemic dynamics, serotype A displayed epidemic dynamics, and SAT2 and SAT3 did not sustain local chains of transmission. Since these results were obtained from the same population at the same time, they highlight important differences in transmission specific to each serotype. They also show that immunity wanes at rates specific to each serotype, which influences patterns of local persistence. Overall, this work shows that viral serotypes can differ significantly in their epidemiological and immunological characteristics. Patterns and processes that drive transmission in endemic settings must consider complex viral dynamics for accurate representation and interpretation.     

On global and local critical points of extended contact process on homogeneous trees

We study spatial stochastic epidemic models called households models. The households models have more than two states at each vertex of a graph in contrast to the contact process. We show that, in the households models on trees, two thresholds of infection rates characterize epidemics. The global critical infection rate is defined by epidemic occurrence. However, some households may be eventually disease-free even for infection rates above the global critical infection rate, in as far as they are smaller than the local critical point. Whether the global one is smaller than the local one depends on the graph and the model. We show that, in the households models, the global one is smaller than the local one on homogeneous trees.     

Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity

Many classical mathematical models for animal infections assume that all infected animals transmit the infection at the same rate, all are equally susceptible, and the course of the infection is the same in all animals. However for some infections there is evidence that seropositives may still transmit the infection, albeit at a lower rate. Animals can also experience more than one episode of the infection although those who have already experienced it have a partial immune resistance. Animals who experience a second or subsequent period of infection may not necessarily exhibit clinical symptoms. The main example discussed is bovine respiratory syncytial virus (BRSV) amongst cattle. We consider simple models with vaccination and homogeneous and proportional mixing between seropositives and seronegatives. We derive an expression for the basic reproduction number, R(o), and perform an equilibrium and stability analysis. We find that it may be possible for there to be two endemic equilibria (one stable and one unstable) for R(o)<1 and in this case at R(o)=1 there is a backwards bifurcation of an unstable endemic equilibrium from the infection-free equilibrium. Then the implications for control strategies are considered. Finally applications to Aujesky's disease (pseudorabies virus) in pigs are discussed.     

Adoptive transfer of CD8alpha+ dendritic cells (DC) isolated from mice infected with Chlamydia muridarum are more potent in inducing protective immunity than CD8alpha- DC

Chlamydial infections are serious public health concerns worldwide. In this study, we examined the role of dendritic cell (DC) subsets in inducing protective immunity against chlamydial infection using an adoptive transfer approach. We found that CD11c+CD8alpha+ (double-positive, DP) DC, compared with CD11c+CD8alpha- (single-positive, SP) DC isolated from infected mice, are more potent inducers of protective immunity. Specifically, mice pretreated with DPDC from infected mice, upon infection with Chlamydia trachomatis mouse pneumonitis (MoPn), experienced significantly less severe body weight loss and in vivo chlamydial growth. Analysis of MoPn-driven cytokine production by immune cells revealed that mice that were treated with DPDC produced significantly higher levels of Th1 (TNF-alpha, IFN-gamma, and IL-12) but lower levels of Th2 (IL-4, IL-5, and IL-13)-related cytokines than the recipients of SPDC following infection challenge. Moreover, DPDC-treated mice displayed significantly higher levels of MoPn-specific IgG2a production and delayed-type hypersensitivity responses compared with SPDC-treated mice. Furthermore, DPDC isolated from infected mice produced higher amounts of IL-12 and IL-10 in vitro in comparison with SPDC. These data indicate that CD8alpha+ DC have a significantly higher capacity in inducing protective immunity compared with CD8alpha- DC, demonstrating the crucial role of DC1-like cells in eliciting protection against C. trachomatis infection.     

Persistence thresholds for phocine distemper virus infection in harbour seal Phoca vitulina metapopulations

1. This paper explores the concept of the critical community size for persistence of infection in wildlife populations. We use as a case study the 1988 epidemic of phocine distemper virus in the North Sea population of harbour seals, Phoca vitulina .
2. We summarize the available data on this epidemic and use it to parameterize a stochastic compartmental model for an infection spreading through a spatial array of patches coupled by nearest-neighbour mixing, with replacement of susceptibles occurring as a discrete annual event.
3. A combination of analytical and simulation techniques is used to show that the high levels of transmission between different seal subpopulations, combined with the small annual birth cohort, act to make persistence of infection impossible in this harbour seal population at realistic population levels. The well known mechanisms by which metapopulation structures may act to promote persistence can be seen to have an effect only at weaker levels of spatial coupling, and higher levels of host recruitment, than those empirically observed.     

Modeling malaria vaccines. II: Population effects of stage-specific malaria vaccines dependent on natural boosting

Population effects of malaria vaccination programs will depend on the stage specificity of the vaccine, its duration of effectiveness, whether it is responsive to natural boosting, the proportion vaccinated, and the preexisting endemic conditions. This paper develops models of infection-blocking (sporozoite), disease-modifying (merozoite), and transmission-blocking (gametic) vaccines. It explores numerically their different effects on prevalence of infection and disease when utilized in different types of immunization programs at various levels of coverage. Simulations show that possible qualitative consequences of malaria vaccination programs include decreased prevalence of infection and disease and decreased prevalence of infection without a corresponding decrease in prevalence of disease. Epidemics, either one-time or cyclical, could occur. These effects could be accompanied by changes in the age distribution of disease. Finally, vaccination could contribute to elimination of transmission. The duration of effectiveness of the malaria vaccine relative to the duration of natural immunity could have important consequences for the unvaccinated. The problem of predicting a threshold for elimination of transmission is discussed.     

Role of dendritic cells in antibody-dependent enhancement of dengue virus infection

Dengue viruses (DV), composed of four distinct serotypes (DV1 to DV4), cause 50 to 100 million infections annually. Durable homotypic immunity follows infection but may predispose to severe subsequent heterotypic infections, a risk conferred in part by the immune response itself. Antibody-dependent enhancement (ADE), a process best described in vitro, is epidemiologically linked to complicated DV infections, especially in Southeast Asia. Here we report for the first time the ADE phenomenon in primary human dendritic cells (DC), early targets of DV infection, and human cell lines bearing Fc receptors. We show that ADE is inversely correlated with surface expression of DC-SIGN (DC-specific intercellular adhesion molecule-3-grabbing nonintegrin) and requires Fc gamma receptor IIa (FcgammaRIIa). Mature DC exhibited ADE, whereas immature DC, expressing higher levels of DC-SIGN and similar FcgammaRIIa levels, did not undergo ADE. ADE results in increased intracellular de novo DV protein synthesis, increased viral RNA production and release, and increased infectivity of the supernatants in mature DC. Interestingly, tumor necrosis factor alpha and interleukin-6 (IL-6), but not IL-10 and gamma interferon, were released in the presence of dengue patient sera but generally only at enhancement titers, suggesting a signaling component of ADE. FcgammaRIIa inhibition with monoclonal antibodies abrogated ADE and associated downstream consequences. DV versatility in entry routes (FcgammaRIIa or DC-SIGN) in mature DC broadens target options and suggests additional ways for DC to contribute to the pathogenesis of severe DV infection. Studying the cellular targets of DV infection and their susceptibility to ADE will aid our understanding of complex disease and contribute to the field of vaccine development.     

Dynamics of acquired immunity boosted by exposure to infection

A mathematical model is presented to describe the dynamics of immunity which can be boosted by reexposure to infection. Immunity is assumed to last until a specified interval of time elapses without an exposure. This assumption is incorporated into a compartmental model as a differential-delay equation. When the model is applied to malaria epidemiology, the prevalence of disease among adults is greatest at intermediate rates of infection. Observed age-prevalence curves have shapes similar to those of the predicted curves. Immunity in an individual is formulated in terms of a stochastic process, and an expression for the average duration of immunity is obtained. The average duration of immunity increases with the rate of exposure, but the presence of mortality (or other kinds of removal) shortens the average duration observed, in analogy with the theoryof competing risks.     

On the transmission of HIV with self-protective behavior and preferred mixing

An epidemic model of HIV transmission with self-protective behavior and preferred mixing is presented. Individuals in the model are assumed to choose their levels of risk behavior by comparing the costs and benefits of self-protective actions. Unlike in models which treat individual risk behavior as exogenously given and fixed, the condition under which an endemic steady state equilibrium exists does not depend on the extent of assortative mixing in the population. Specifically, a unique endemic equilibrium exists when the basic reproductive number of the disease, which is given in the model by the expected number of secondary infections caused by an infected individual in the absence of any self-protection, is strictly greater than one. Otherwise, the disease-free equilibrium is the only steady state equilibrium. With respect to changes in contact patterns, it is shown that, if the degree of preferred mixing is increased, the disease prevalence can decrease in the high-risk subpopulation consisting of individuals who are more likely to engage in unsafe practices. The situation is reversed for the low-risk subpopulation, which is composed of individuals who are less willing to engage in risky practices, so that increasing the likelihood of mixing with members of one's own group may increase the prevalence level within the low-risk subpopulation.     

Predicting species distributions in poorly-studied landscapes

Conservationists are increasingly relying on distribution models to predict where species are likely to occur, especially in poorly-surveyed but biodiverse areas. Modeling is challenging in these cases because locality data necessary for model formation are often scarce and spatially imprecise. To identify methods best suited to modeling in these conditions, we compared the success of three algorithms (Maxent, Mahalanobis Typicalities and Random Forests) at predicting distributions of eight bird and eight mammal species endemic to the eastern slopes of the central Andes. We selected study species to have a range of locality sample sizes representative of the data available for endemic species of this region and also that vary in their distribution characteristics. We found that for species that are known from moderate numbers (N = 38–94) of localities, the three methods performed similarly for species with restricted distributions but Maxent and Random Forests yielded better results for species with wider distributions. For species with small numbers of sample localities (N = 5–21), Maxent produced the most consistently successful results, followed by Random Forests and then Mahalanobis Typicalities. Because evaluation statistics for models derived from few localities can be suspect due to the poor spatial representation of the evaluation data, we corroborated these results with review by scientists familiar with the species in the field. Overall, Maxent appears to be the most capable method for modeling distributions of Andean bird and mammal species because of the consistency of results in varying conditions, although the other methods have strengths in certain situations. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection

In stochastic modelling of infectious spread, it is often assumed that infection confers permanent immunity, a susceptible-infective-removed (SIR) model. We show how results concerning long-term (endemic) behaviour may be extended to a susceptible-infective-removed-susceptible (SIRS) model, in which immunity is temporary. Since the full SIRS model with demography is rather intractable, we also consider two simpler models: the susceptible-infective-susceptible (SIS) model with demography, in which there is no immunity; and the SIRS model in a closed population. For each model, we first analyse a deterministic model, then approximate the quasi-stationary distribution (equilibrium distribution conditional upon non-extinction of infection) using a moment closure technique. We look in particular at the effect of the immune period upon infection prevalence and upon time to fade-out of infection. Our main findings are that a shorter average immune period leads to higher infection prevalence in quasi-stationarity, and to longer persistence of infection in the population.